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What is a Sector (part of a circle)?
Grade Level:
Class 2
All STEM domains, Finance, Economics, Data Science, AI, Physics, Chemistry
Definition
What is it?
A sector is like a slice of a circular pizza or cake. It is a part of a circle enclosed by two radii (lines from the center to the edge) and the arc connecting them.
Simple Example
Quick Example
Imagine you have a round roti. If you cut a piece out from the center to the edge, like a V-shape, that piece is a sector. It has two straight edges meeting at the center and one curved edge.
Worked Example
Step-by-Step
Let's find the area of a sector of a circle with a radius of 7 cm and a central angle of 90 degrees.
1. Understand the formula: Area of a sector = (theta / 360) * pi * r^2, where theta is the central angle and r is the radius.
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2. Identify the given values: Radius (r) = 7 cm, Central angle (theta) = 90 degrees, pi = 22/7.
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3. Substitute the values into the formula: Area = (90 / 360) * (22/7) * 7^2.
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4. Simplify the fraction: 90 / 360 = 1/4.
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5. Calculate r^2: 7^2 = 49.
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6. Substitute back: Area = (1/4) * (22/7) * 49.
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7. Perform multiplication: Area = (1/4) * (22 * 7) = (1/4) * 154.
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8. Final calculation: Area = 154 / 4 = 38.5 square cm.
Answer: The area of the sector is 38.5 square cm.
Why It Matters
Understanding sectors helps us calculate areas of curved shapes, which is crucial in fields like architecture for designing domes or in engineering for gears. It's also used in data science to create pie charts, helping visualize data clearly.
Common Mistakes
MISTAKE: Confusing a sector with a segment. | CORRECTION: A sector is formed by two radii and an arc, like a pizza slice. A segment is formed by a chord and an arc, like the crust of a pizza cut straight across.
MISTAKE: Using the full circle's area formula (pi*r^2) without considering the angle. | CORRECTION: Remember to multiply the full circle area by the fraction (central angle / 360) to get the area of only the specific sector.
MISTAKE: Forgetting to use the correct units for area (e.g., cm instead of cm^2). | CORRECTION: Area is always measured in square units (like cm^2, m^2). Lengths like radius and arc length are in linear units (cm, m).
Practice Questions
Try It Yourself
QUESTION: A circular park has a sprinkler that waters a sector with a radius of 10 meters and a central angle of 60 degrees. What fraction of the full circle is watered? | ANSWER: 1/6
QUESTION: Find the area of a sector of a circle with a radius of 14 cm and a central angle of 45 degrees. (Use pi = 22/7) | ANSWER: 77 square cm
QUESTION: A pie chart shows that the 'Cricket' section has a central angle of 120 degrees. If the total area of the pie chart is 90 square cm, what is the area of the 'Cricket' section? | ANSWER: 30 square cm
MCQ
Quick Quiz
Which of these everyday items represents a sector?
A full round chapati
A piece of a pizza slice
A straight ruler
A square photo frame
The Correct Answer Is:
B
A piece of a pizza slice is cut from the center to the edge, just like a sector. A full chapati is a whole circle, a ruler is a line, and a photo frame is a square.
Real World Connection
In the Real World
Pie charts, often seen in news reports about election results or company profits, are made up of different sectors. Each sector represents a part of the whole, like how much a political party scored or how much profit a company made from different products.
Key Vocabulary
Key Terms
RADIUS: A line segment from the center of a circle to any point on its circumference | ARC: A continuous part of the circumference of a circle | CENTRAL ANGLE: The angle formed by two radii at the center of a circle | CIRCUMFERENCE: The perimeter or boundary of a circle
What's Next
What to Learn Next
Now that you understand what a sector is, you can learn about calculating its arc length and perimeter. This will help you solve more complex problems involving parts of circles and their measurements.
